Rainbow Team Perfect clear the specified stage

Building Cannons on the top row to take out the first wave of enemies
and then build a Arrow Tower on the tree closest to the base and upgrade it till you get the yellow flag.
Wave 4 and 5 you are going to need to buy the Laser Towers so make sure to save your gems. Build the tower at the the start of wave 5 and upgrade it till you get the red flag. Build two more Laser Towers and upgrade one to red flag and the other to yellow.
Build Fire Towers at around wave 11 or 12 and upgrade them to purple flag as quickly as possible. I suggest placing these near the entrance.
Once you've achieved this immediately go back to working on the Laser Towers get them to purple as soon as you can.
After getting them to purple save up and build as many Fire Towers as you can getting them to purple every time. Be sure to try and place more of these towers on the left side then the right.
Before you the boss arrives on wave 20 replace all the arrow towers with either cannons or fire towers and upgrade the replaced towers as much as possible.

Get Medieval Clear the specified stage using only arrows and cannons

This is a pretty hard trophy to achieve due to the fact that only Arrows and Cannons can be used. It also helps a lot knowing you can use mines. Play it like any other level and use mines when needed. I recommend using at least one for 19 since the enemies are resistant to Arrows and Cannons.

Tower Collector Clear the specified stage by building every type of tower

Another pretty easy trophy simply save all your gems for unlocking towers instead of using them for quick upgrades. Just look at the enemies coming in the next wave and you will know what tower to unlock. Start off using Arrows and Cannons and be sure to place an Ice Tower because it will be needed so make sure it can reach the enemies.

No Crossing Clear the specified stage without crossing the bridge

Build Arrow and Cannon Towers at the entrance of the bridge. Be sure to save your gems for Laser Towers and only build Anti-Air Guns when needed which you should only have a max of 3 if even that. Right when you unlock the Laser Tower build one in the group of trees by the bridge and 2 more up top by your base. This should be able to hold the next couple of waves that come through. When possible build 2 Fire Towers at the entrance of the bridge and upgrade to red flag immediately.

Green Flag Clear the specified stage without a tower ever upgrading

This is a slightly hard trophy to get you just have to pay attention. Check all trees for gems and coins. Be sure to constantly be checking towers because every enemy they kill will give them XP. When you get enough gems buy interests and keep doing so until the end.

Picky Eater Clear the specified stage without using the fire tower

Gem Hoarder Clear the specified stage with 25 or more gems

This level is a little more complicated then the rest. Due to the fact that there are only 22 gems on the level and 1 hidden in the trees. So how do you get the last 2 gems you ask. Good question you can either upgrade 2 of your towers to purple and sell them which both will give you a gem or simply upgrade 1 tower to black and sell it which will give you 2 gems.

Scrooge Clear the specified stage with a balance of 10,000 worth of coins or more

Start out building a few at the entrance and a few by your base, then simply upgrade the towers and only make new ones when needed. Sell towers you don't need and be sure to put all your gems towards buying interest.

Deep Impact Destroy 15 enemies at once with the Mortar Tower

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Encore : Black Flag Without selling any towers, clear the specified stage with all towers upgraded to black

Go to the right of your base and simply place 4 Cannons, 3 Arrow, and 2 Anti-Air Towers. Upgrade them all to red flag and then after building your first Anti-Air Tower make sure you have 7 gems to upgrade it to black flag right away. Once all the flying enemies are dead you will get a lot of coins and gems. Sadly you will lose 1 or 2 from your base but you will still receive the trophy.